document.write( "Question 316992: Factor:
\n" ); document.write( "3x^2+2xy-16y^2 \n" ); document.write( "

Algebra.Com's Answer #226918 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"3x%5E2%2B2xy-16y%5E2\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"-16y%5E2\"$ where the coefficients are 3 and -16 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -16 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient 2? Let's list all of the factors of -48:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -48:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -48\r
\n" ); document.write( "\n" ); document.write( "(1)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(16)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(8)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2\r
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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and 8 add up to 2 and multiply to -48\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3x%5E2%2B2xy-16y%5E2\"$ , replace $\"2xy\"$ with $\"-6xy%2B8xy\"$ (notice $\"-6xy%2B8xy\"$ adds up to $\"2xy\"$ . So it is equivalent to $\"2xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-6xy%2B8xy%29%2B-16y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3x%5E2-6xy%2B8xy-16y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-6xy%29%2B%288xy-16y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x-2y%29%2B8y%28x-2y%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"8y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B8y%29%28x-2y%29\"$ Since we have a common term of $\"x-2y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-6xy%2B8xy-16y%5E2\"$ factors to $\"%283x%2B8y%29%28x-2y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3x%5E2%2B2xy-16y%5E2\"$ factors to $\"%283x%2B8y%29%28x-2y%29\"$ (since $\"3x%5E2%2B2xy-16y%5E2\"$ is equivalent to $\"3x%5E2-6xy%2B8xy-16y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B2xy-16y%5E2\"$ factors to $\"%283x%2B8y%29%28x-2y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2%2B2xy-16y%5E2=%283x%2B8y%29%28x-2y%29\"$
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